Multiplication of fourier series

Say you have two functions, F(x,y), and G(x,y), and you want to expand them in finite fourier series. Let their coefficients be designated as F_ij and G_ij. When you multiply the two functions, you get X=FG, and this should also have its own fourier series, call its components X_mn. What is the relation between F_ij, G_ij, and X_mn?

I was hoping you had something like X_ij = F_ij G_ij, but I've been looking at this for a little while and it seems you don't have any nice relation like that.

This, by the way, resembles the convolution theorem of the Fourier Transform, and actually the above operation between two sequences (in this special case they are two dimensional) is a discrete convolution.